Carrier Multiplication in Transition Metal Dichalcogenides Beyond Threshold Limit

Abstract Carrier multiplication (CM), multiexciton generation by absorbing a single photon, enables disruptive improvements in photovoltaic conversion efficiency. However, energy conservation constrains the threshold energy to at least twice bandgap (2Eg). Here, a below threshold limit CM in monolayer transition metal dichalcogenides (TMDCs) is reported. Surprisingly, CM is observed with excitation energy of only 1.75 Eg due to lattice vibrations. Electron–phonon coupling (EPC) results in significant changes in electronic structures, which favors CM. Indeed, the strongest EPC in monolayer MoS2 leads to the most efficient CM among the studied TMDCs. For practical applications, chalcogen vacancies can further lower the threshold by introducing defect states within bandgap. In particular, for monolayer WS2, CM occurs with excitation energy as low as 1.51 Eg. The results identify TMDCs as attractive candidate materials for efficient optoelectronic devices with the advantages of high photoconductivity and efficient CM.

molybdenum to tungsten owing to the decrease in electronegativity and increase of ionic radii. Excitation Dynamics in monolayer MoTe 2 To investigate CM in monolayer MoTe 2 , electron-hole pairs with energy of 2E g are created, in which hole excess energy ∆E h is 0.86E g and electron excess energy ∆E e is 0.14E g (Fig-ure S1(c)). As discussed in the main text, the phonon-induced reduction of bandgap drives the onset of CM, in which excited holes have sufficient energy to scatter additional electronhole pairs in the system. Compared with experimental observation, where the CM conversion efficiency η CM can be as high as 94∼99%, 1,2 the carrier generation quantum yield (QY) in our simulations is inefficient. The origin of the relatively low CM conversion efficiency in our calculations is as follows. Firstly, twelve electron-hole pairs are excited in monolayer MoTe 2 , with a concentration of 3.58×10 −14 cm −2 which is larger than that in experiment work. Thus, an efficient Auger recombination is resulted. As shown in Figure S3(a)-(b), partial excited holes and electrons populate higher energy states, which is the fingerprint of Auger process.
This decay dynamics can be fitted by exponential decay, with a lifetime of ∼1.98 ps. This can be attributed to Auger trapping free carriers which is detected on 1-2 ps time scale in TMDCs. [3][4][5][6] Furthermore, excitation dynamics of monolayer MoTe 2 with one electron-hole pair is simulated, as shown in Figure S4. The maximum carrier generation QY equals to 2, and conversion efficiency η CM reaches up to 100% which is consistent with experimental observations. 1,2 In addition, surface/defect trapping which effectively competes with Auger process also make contributions to the higher conversion efficiency in experiments. Apart from Auger recombination, monolayer crystal structure accelerates charge carrier nonradiative recombination. Figure S3(a)-(b) present that excited hole orbitals partially occupy conduction band states while excited electrons populate valence band states, corresponding to the nonradiative recombination. Generally, the charge carrier nonradiative recombination rate in monolayer TMDCs is on the order of several hundreds of picoseconds. 7,8 For further analysis the role of EPC in CM process, carriers are excited in monolayer MoTe 2 with energy of 2E g at 77 K, 300 K and 500 K, respectively. Excitations with two different excess energy distribution are simulated (∆E e/h =0.14/0.86E g and ∆E e/h =0.50/0.50E g ).
As discussed in the main text, for asymmetric electron-hole pair, CM phenomenon is observed at all three temperatures. It is found that increasing temperature has a positive effect  This results in a narrower bandgap and consequently more efficient carrier generation QY.
As shown in Figure S5 and Figure 1, from 77 K to 500 K, the reduction of bandgap increasing from 28% to 64%. If keep increasing temperature, semicondutor-to-metal transitions may eventually kick in. In the case of carriers with excess energies of ∆E e/h =0.5/0.5E g , CM phenomenon is only observed in 500 K among the three cases. Consistently, Figure S6(a) shows that the electron excess energy ∆E e (t) is beyond E g (t) at about 100 fs. In contrast, E g (t) is always larger than ∆E e/h (t) after excitation in the case of 77 K and 300 K. Therefore, it is possible to break the threshold limit for CM via coupling to phonons.  To further demonstrate the role of bandgap in CM process, we apply strain along the lattice parameter directions in monolayer MoTe 2 , with a range of -3% ∼ +3%. The inset of Figure S12 shows that tensile and compressive strains both narrow the bandgap of MoTe 2 which is expected to enhance QY of CM. As shown in Figure S12, +3% strain improve QY of CM slightly, with a value of ∼1.08, while QY in the case with -3% strain gives rise up to ∼1.12, increasing around 30%. This results indicate that the narrower bandgap of monolayer TMDC indeed have positive effect in improving CM performance, which can be manipulated by nuclear vibrations.

Chalcogen vacancy defect
To optimize the electronic structures of TMDCs to enhance their performance for photovoltaic applications, a sulfur vacancy is introduced to the 3 √ 3 × 3 √ 3 supercell of monolayer WS 2 ( Figure S1(b)), corresponding to a defect concentration of 3.66× 10 13 cm −2 . As shown in Figure S13, the calculated bandgap of pristine monolayer WS 2 is 1.95 eV, which is con-